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Thread: Create a 3x3 grid of unique integers squared, each row, column, diag. must equal N

  1. #1
    New Member chris84567's Avatar
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    Create a 3x3 grid of unique integers squared, each row, column, diag. must equal N

    Not sure if this is the correct category, if its not sorry.

    If you create a 3x3 grid of unique integers squared, each row, column and diagonal has to be equal to N.
    What is one solution to this problem and what is a general solution to this problem?

    n
    a2 b2 c2 n
    d2 e2 f2 n
    g2 h2 i2 n
    n n n n

    Using systems of equtions I was able to find that e2= 1/3n.

    a2 +b2 +c2 =n
    -(a2 +e2 +i2 =n)
    -(b2 +e2 +h2 =n)
    -(c2 +e2 +g2 =n)
    g2 +h2 +i2 =n
    =
    -3e2 =n

    After this I have been able to write everything in terms of a, c and n but don't know where to go next.


    n
    a2 b2 c2 n
    d2 1n/3 f2 n
    g2 h2 i2 n
    n n n n


    Thanks for any help you can offer.
    Last edited by chris84567; 01-24-2018 at 12:07 PM. Reason: Duplicate information

  2. #2
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    First thing to notice is that you have 9 unknowns, but only 8 equations. So you won't, in general, have a single unique solution. You'll probably have infinitely-many solutions.

    What you did is fine. However, each equation can be used to reduce the number of variables, one at a time.

    a^2 + b^2 + c^2 = n

    a^2 = n - (b^2 + c^2)

    In principle you could just substitute this in everywhere that you see a^2, and that variable will have been eliminated. Repeat.

  3. #3
    New Member chris84567's Avatar
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    Quote Originally Posted by j-astron View Post
    In principle you could just substitute this in everywhere that you see a^2, and that variable will have been eliminated. Repeat.
    I have substituted and got every thing in terms of a and c.

    Im in school right now but when I get home I'll share what I got.

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